Finite-dimensional Reduction Method Research Articles

Solutions with multiple peaks for nonlinear Kirchhoff equations on R^3

In this paper, we mainly investigate the following nonlinear Kirchhoff equation
 \(-\left(\epsilon^2 a+\epsilon b\int_{\mathbb{R}^3}|\nabla u|^2\right)\Delta u +u =Q(x)u^{q-1}\), \(u>0\), \(x\in\mathbb{R}^{3}\),
 \(u\to 0\), as \(|x|\to +\infty\),
 
 where \(a,b>0\) are constants, \(2<q<6\), and \(\epsilon>0\) is a parameter. Under some suitable assumptions on the function \(Q(x)\), we obtain that the equation above has positive multi-peak solutions concentrating at a critical point of \(Q(x)\) for \(\epsilon>0\) sufficiently small, by using the finite dimensional reduction method. Different from the local Schrödinger problem, here the corresponding limit problem is a system. Moreover, the nonlocal term brings some new difficulties which involve some technical and complicated estimates.

Existence and local uniqueness of normalized peak solutions for a Schrödinger-Newton system

In this paper, we investigate the existence and local uniqueness of normalized peak solutions for a Schr\"odinger-Newton system under the assumption that the trapping potential is degenerate and has non-isolated critical points. First we investigate the existence and local uniqueness of normalized single-peak solutions for the Schr\"odinger-Newton system. Precisely, we give the precise description of the chemical potential $\mu$ and the attractive interaction $a$. Then we apply the finite dimensional reduction method to obtain the existence of single-peak solutions. Furthermore, using various local Pohozaev identities, blow-up analysis and the maximum principle, we prove the local uniqueness of single-peak solutions by precise analysis of the concentrated points and the Lagrange multiplier. Finally, we also prove the nonexistence of multi-peak solutions for the Schr\"odinger-Newton system, which is markedly different from the corresponding Schr\"odinger equation. The nonlocal term results in this difference. The main difficulties come from the estimates on Lagrange multiplier, the different degenerate rates along different directions at the critical point of $P(x)$ and some complicated estimates involved by the nonlocal term. To our best knowledge, it may be the first time to study the existence and local uniqueness of solutions with prescribed $L^{2}$-norm for the Schr\"odinger-Newton system.

Existence and multiplicity of rotating periodic solutions for Hamiltonian systems with a general twist condition

In this paper, we consider rotating periodic solutions of the Hamiltonian systemx˙=JH′(t,x),x∈R2N, having the form x(t+T)=Qx(t),∀t∈R, for some T>0 and a symplectic orthogonal matrix Q. We study the system under a general twist condition: the nonlinear term H′(t,x) is required to be of linear growth but not necessarily to be asymptotically linear at infinity. The twist is reflected in the difference of the generalized Morse index at the origin and at infinity. By combining a finite dimensional reduction method, Morse theory and minimax principle, we establish the existence and multiplicity of nontrivial rotating periodic solutions.

Local uniqueness of concentrated solutions and some applications on nonlinear Schrödinger equations with very degenerate potentials

We revisit the following nonlinear Schrödinger equation−ε2Δu+V(x)u=up−1,u>0,u∈H1(RN), where ε>0 is a small parameter, N≥2 and 2<p<2⁎. We obtain a more accurate location for the concentrated points, the existence and the local uniqueness for positive k-peak solutions when V(x) possesses non-isolated critical points by using the modified finite dimensional reduction method based on local Pohozaev identities. Moreover, for several special potentials, with its critical point set being a low-dimensional ellipsoid, or a part of hyperboloid of one sheet or two sheets, we obtain the number and symmetry of k-peak solutions by using local uniqueness of concentrated solutions. Here the main difficulty comes from the different degenerate rate along different directions at the critical points of V(x).

Excited states for two-component Bose-Einstein condensates in dimension two

We consider the following two-component Bose-Einstein condensates (BEC) by investigating the associated L2-critical problem{−Δu1+V1(x)u1=a1u13+βu1u22+μu1inR2,−Δu2+V2(x)u2=a2u23+βu12u2+μu2inR2, with the constraint ∫R2(u12+u22)dx=1.Ground states and excited states are typically described in BEC phenomenon. In this paper, we prove the existence of the general exited states by use of the finite dimensional reduction method combined with the local Pohozaev identities. Precisely, we construct vector peak solutions as a1+a2−2βa1a2−β2→1ka⁎, where a1,a2>0,β∈(−a1a2,min⁡{a1,a2})∪(max⁡{a1,a2},+∞), k is a positive integer, a⁎=∫R2W2 and W is the unique positive solution of −Δw+w=w3 in R2. Our results generalize those obtained in [9], where Guo-Li-Wei-Zeng proved the existence of the ground states for fixed 0<a1,a2≤a⁎ and β>0 by variational methods.Compared with the single equation, the difficulties in the study of BEC systems come from the interspecies interaction between the components, which requires subtle estimates.

On the singularly perturbation fractional Kirchhoff equations: Critical case

Abstract This article deals with the following fractional Kirchhoff problem with critical exponent a + b ∫ R N ∣ ( − Δ ) s 2 u ∣ 2 d x ( − Δ ) s u = ( 1 + ε K ( x ) ) u 2 s ∗ − 1 , in R N , \left(a+b\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}| {\left(-\Delta )}^{\tfrac{s}{2}}u\hspace{-0.25em}{| }^{2}{\rm{d}}x\right){\left(-\Delta )}^{s}u=\left(1+\varepsilon K\left(x)){u}^{{2}_{s}^{\ast }-1},\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N}, where a , b &gt; 0 a,b\gt 0 are given constants, ε \varepsilon is a small parameter, 2 s ∗ = 2 N N − 2 s {2}_{s}^{\ast }=\frac{2N}{N-2s} with 0 &lt; s &lt; 1 0\lt s\lt 1 and N ≥ 4 s N\ge 4s . We first prove the nondegeneracy of positive solutions when ε = 0 \varepsilon =0 . In particular, we prove that uniqueness breaks down for dimensions N &gt; 4 s N\gt 4s , i.e., we show that there exist two nondegenerate positive solutions which seem to be completely different from the result of the fractional Schrödinger equation or the low-dimensional fractional Kirchhoff equation. Using the finite-dimensional reduction method and perturbed arguments, we also obtain the existence of positive solutions to the singular perturbation problems for ε \varepsilon small.

Existence of multi-bump solutions for a system with critical exponent in [formula omitted

We consider the following system with critical exponent in RN:{−Δu=K1(y)u2⁎−1+p2⁎V(y)up−1vq in RN,−Δv=K2(y)v2⁎−1+q2⁎V(y)upvq−1 in RN,u,v>0,y∈RN, where N≥5, p,q>1 and p+q=2⁎=2NN−2. Using finite dimensional reduction method, we prove the existence of multi-bump solutions. Their bumps can be placed on arbitrarily many or even infinitely many lattice points in RN. Since p<2 or q<2, we introduce two new norms to avoid singularity.

Solutions of the Yamabe Equation by Lyapunov–Schmidt Reduction

Given any closed Riemannian manifold (M, g) we use the Lyapunov–Schmidt finite-dimensional reduction method and the classical Morse and Lusternick–Schnirelmann theories to prove multiplicity results for positive solutions of a subcritical Yamabe type equation on (M, g). If (N, h) is a closed Riemannian manifold of constant positive scalar curvature we obtain multiplicity results for the Yamabe equation on the Riemannian product \((M\times N , g + \varepsilon ^2 h )\), for \(\varepsilon >0\) small. For example, if M is a closed Riemann surface of genus \(\mathbf{g}\) and \((N,h) = (S^2 , g_0)\) is the round 2-sphere, we prove that for \(\varepsilon >0\) small enough and a generic metric g on M, the Yamabe equation on \((M\times S^2 , g + \varepsilon ^2 g_0 )\) has at least \(2 + 2 \mathbf{g}\) solutions.

Large number of bubble solutions for a fractional elliptic equation with almost critical exponents

This paper deals with the following non-linear equation with a fractional Laplacian operator and almost critical exponents: \[ (-\Delta)^{s} u=K(|y'|,y'')u^{({N+2s})/(N-2s)\pm\epsilon},\quad u &gt; 0,\quad u\in D^{1,s}(\mathbb{R}^{N}), \] where N ⩾ 4, 0 &lt; s &lt; 1, (y′, y″) ∈ ℝ2 × ℝN−2, ε &gt; 0 is a small parameter and K(y) is non-negative and bounded. Under some suitable assumptions of the potential function K(r, y″), we will use the finite-dimensional reduction method and some local Pohozaev identities to prove that the above problem has a large number of bubble solutions. The concentration points of the bubble solutions include a saddle point of K(y). Moreover, the functional energies of these solutions are in the order $\epsilon ^{-(({N-2s-2})/({(N-2s)^2})}$.

Infinitely Many Solitary Waves Due to the Second-Harmonic Generation in Quadratic Media

In this paper, we consider the following coupled Schrodinger system with χ(2) nonlinearities $\left\{ \begin{array}{l} - \Delta {u_1} + {V_1}\left( x \right){u_1} = \alpha {u_1}{u_2},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x \in {^N}, \\ - \Delta {u_2} + {V_2}\left( x \right){u_2} = \frac{\alpha }{2}u_1^2 + \beta u_2^2,\,\,\,\,\,\,\,\,\,\,\,\,\,x \in {^N}, \\ \end{array} \right.$ which arises from second-harmonic generation in quadratic media. Here V1(x) and V2(x) are radially positive functions, 2 ≤ N 0 and α > β. Assume that the potential functions V1(x) and V2(x) satisfy some algebraic decay at infinity. Applying the finite dimensional reduction method, we construct an unbounded sequence of non-radial vector solutions of synchronized type.

Solutions for Fractional Schrödinger Equation Involving Critical Exponent via Local Pohozaev Identities

Abstract We consider the following fractional Schrödinger equation involving critical exponent: { ( - Δ ) s ⁢ u + V ⁢ ( y ) ⁢ u = u 2 s * - 1 in ⁢ ℝ N , u &gt; 0 , y ∈ ℝ N , \left\{\begin{aligned} &amp;\displaystyle(-\Delta)^{s}u+V(y)u=u^{2^{*}_{s}-1}&amp;&amp;% \displaystyle\text{in }\mathbb{R}^{N},\\ &amp;\displaystyle u&gt;0,&amp;&amp;\displaystyle y\in\mathbb{R}^{N},\end{aligned}\right. where N ≥ 3 {N\geq 3} and 2 s * = 2 ⁢ N N - 2 ⁢ s {2^{*}_{s}=\frac{2N}{N-2s}} is the critical Sobolev exponent. Under some suitable assumptions of the potential function V ⁢ ( y ) {V(y)} , by using a finite-dimensional reduction method, combined with various local Pohazaev identities, we prove the existence of infinitely many solutions. Due to the nonlocality of the fractional Laplacian operator, we need to study the corresponding harmonic extension problem.

Multi-peak Positive Solutions of a Nonlinear Schrödinger–Newton Type System

Abstract In this paper, we study the following nonlinear Schrödinger–Newton type system: { - ϵ 2 ⁢ Δ ⁢ u + u - Φ ⁢ ( x ) ⁢ u = Q ⁢ ( x ) ⁢ | u | ⁢ u , x ∈ ℝ 3 , - ϵ 2 ⁢ Δ ⁢ Φ = u 2 , x ∈ ℝ 3 , \left\{\begin{aligned} &amp;\displaystyle{-}\epsilon^{2}\Delta u+u-\Phi(x)u=Q(x)|u% |u,&amp;&amp;\displaystyle x\in\mathbb{R}^{3},\\ &amp;\displaystyle{-}\epsilon^{2}\Delta\Phi=u^{2},&amp;&amp;\displaystyle x\in\mathbb{R}^{% 3},\end{aligned}\right. where ϵ &gt; 0 {\epsilon&gt;0} and Q ⁢ ( x ) {Q(x)} is a positive bounded continuous potential on ℝ 3 {\mathbb{R}^{3}} satisfying some suitable conditions. By applying the finite-dimensional reduction method, we prove that for any positive integer k, the system has a positive solution with k-peaks concentrating near a strict local minimum point x 0 {x_{0}} of Q ⁢ ( x ) {Q(x)} in ℝ 3 {\mathbb{R}^{3}} , provided that ϵ &gt; 0 {\epsilon&gt;0} is sufficiently small.

A perturbed Kirchhoff problem with critical exponent

ABSTRACT In this paper, we consider a class of Kirchhoff problems as follows (1) where a, b>0 are given constants, ϵ is a small parameter and . We first prove the nondegeneracy of positive solutions to Equation (1) when . Then, as an application, using a perturbed argument and finite-dimensional reduction method, we prove the existence of positive solutions of Equation (1) for ϵ small.

A pointwise finite-dimensional reduction method for a fully coupled system of Einstein–Lichnerowicz type

We construct non-compactness examples for the fully coupled Einstein–Lichnerowicz constraint system in the focusing case. The construction is obtained by combining pointwise a priori asymptotic analysis techniques, finite-dimensional reductions and a fixed-point argument. More precisely, we perform a fixed-point procedure on the remainders of the expected blow-up decomposition. The argument consists of an involved finite-dimensional reduction coupled with a ping-pong method. To overcome the non-variational structure of the system, we work with remainders which belong to strong function spaces and not merely to energy spaces. Performing both the ping-pong argument and the finite-dimensional reduction therefore heavily relies on the a priori pointwise asymptotic techniques of the [Formula: see text] theory.

Degree of locally condensing perturbations of Fredholm maps with positive index and applications

We give a construction of the topological degree for locally condensing perturbations of C1-Fredholm maps of positive index with values in the GLc-framed cobordism group. The construction of the degree extends constructions of the degree for Fredholm maps of positive index and also its finite-dimensional, compact, and compactly restricted perturbations. It based on the finite-dimensional reduction method. We apply the constructed degree to prove the existence theorem for a system of ordinary differential equations with Hopf's boundary conditions.

Prescribed Gauss curvature problem on singular surfaces

We study the existence of at least one conformal metric of prescribed Gaussian curvature on a closed surface $$\Sigma $$ admitting conical singularities of orders $$\alpha _i$$ ’s at points $$p_i$$ ’s. In particular, we are concerned with the case where the prescribed Gaussian curvature is sign-changing. Such a geometrical problem reduces to solving a singular Liouville equation. By employing a min–max scheme jointly with a finite dimensional reduction method, we deduce new perturbative results providing existence when the quantity $$\chi (\Sigma )+\sum _i \alpha _i$$ approaches a positive even integer, where $$\chi (\Sigma )$$ is the Euler characteristic of the surface $$\Sigma $$ .

A pointwise finite-dimensional reduction method for Einstein–Lichnerowicz-type systems: The six-dimensional case

We construct blowing-up solutions for smooth perturbations of a focusing Einstein–Lichnerowicz system on a closed 6-dimensional manifold. Our construction combines asymptotic analysis techniques with a finite-dimensional reduction approach, following the scheme of proof developed in Premoselli (2016). The blow-up examples constructed here highlight in particular the role of the coupling field in dimension 6.

Some elliptic system and reduction method

We get a theorem which shows the existence of at least three solutions for some elliptic system with Dirichlet boundary condition. We obtain this result by using the finite dimensional reduction method for the dimension of the system which reduces the infinite dimensional problem to the finite dimensional one. We also use critical point theory on the reduced finite dimensional subspace.

Oriented Degree of Fredholm Maps: Finite-Dimensional Reduction Method

The complete oriented degree theory for proper nonlinear Fredholm zero-index maps and their compact perturbations is constructed by means of a finite-dimensional reduction method. Applications to solvability and bifurcations of boundary-value problems for nonlinear elliptic equations are provided.

DIRICHLET BOUNDARY VALUE PROBLEM FOR A CLASS OF THE ELLIPTIC SYSTEM

We get a theorem which shows the existence of at least three solutions for some elliptic system with Dirichlet boundary condition. We obtain this result by using the finite dimensional reduction method which reduces the infinite dimensional problem to the finite dimensional one. We also use the critical point theory on the reduced finite dimensioal subspace.